Theorem rexv 2730

Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
rexv	|- ( E. x e. _V ph <-> E. x ph )

Proof of Theorem rexv

Step	Hyp	Ref	Expression
1		df-rex 2441	. 2 |- ( E. x e. _V ph <-> E. x ( x e. _V /\ ph ) )
2		vex 2715	. . . 4 |- x e. _V
3	2	biantrur 301	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	exbii 1585	. 2 |- ( E. x ph <-> E. x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 186	1 |- ( E. x e. _V ph <-> E. x ph )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1472   e. wcel 2128   E.wrex 2436   _Vcvv 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-rex 2441  df-v 2714

This theorem is referenced by:  rexcom4  2735  spesbc  3022  abnex  4407  dfco2  5085  dfco2a  5086